Two exactly Coulomb interacting electrons in a finite quantum wire and an external magnetic field.

Transcription

1 University of Iceland Computational physics Two exactly Coulomb interacting electrons in a finite quantum wire and an external magnetic field. Author: Ólafur Jónasson Supervisor: Viðar Guðmundsson January 19, 2010

2 Contents 1 Introduction 1 2 The theory and program The many-particle basis and Fermi operators Making the MES basis Calculating the matrix elements Calculating the charge density Bottlenecks in calculations The program Convergence Effects of limited MES basis Effects of limited grid Choosing a suitable η Conclusion on convergence Physical results Energy spectrum General behavior Effects of magnetic field on lowest states Energy splitting Charge density Conclusions 19 i

3 1 Introduction The system I m investigating consists of a GaAs-wire (200nm long) in a constant homogeneous magnetic field. I modified a program which calculated the eigenvalues and eigenvectors of the Hamiltonian of the system without Coulomb interaction. I modified the program in such a way that the calculations include the Coulomb interaction for two or three electrons. Without the interaction the system is not physically interesting because the Coulomb interaction can drastically change quantities of interest. In this thesis I will explain the methods and results of my modification to the program. My approach to the problem will be similar to that in the non-interacting case, but obviously, some things need to be done differently. I will use second quantization reformulation of the Schrödinger equation [1]. One of the advantages of second quantization for these calculations is that it incorporates the Fermi particle statistics so there is no need for the computationally/memory expensive task of dealing directly with antisymmetric products of single-particle wave functions or many-particle wave functions. The calculations will be done exactly, that is, the only approximation is the truncation of the single-electron states (SES) and many-electron states (MES) basis, the finite grid used to store the SES wave functions and the error in the numerical integration. How these errors effect the accuracy of the calculations will be investigated by varying the size of the basis, grid and other external parameters. 2 The theory and program 2.1 The many-particle basis and Fermi operators For the calculations we need a MES basis. We will construct it from the noninteracting single-electron states. The MES are time-independent abstract state vectors n 1 n 2 n 3 n = Ψ 1 Ψ 2 Ψ 3 Ψ, (2.1) where n 1 denotes how many-electrons are in SES 1, n 2 how many in SES 2 etc. and Ψ i are single-electron states. Obviously the MES basis fulfills the orthogonality and completeness conditions. We are only considering fermions (electrons) so n i = 0 or n i = 1 (the Pauli inclusion principle). We will also use the Fermi creation and destruction operators ĉ k and ĉ k with the properties 1

4 ĉ k c k n k = n k n k { n = 0, 1 (2.2) ( 1) γ k 0 if nk = 1 ĉ k n k = 0 if n k = 0 { (2.3) ĉ k n 0 if n k = 1 k = ( 1) γ k 1 if nk = 0 (2.4) where k 1 γ k = n i. (2.5) Using (2.1) - (2.5) the Hamiltonian of the system has been transformed into: Ĥ = a i=1 E a ĉ rĉ s + 1 ĉ 2 aĉ b ab V coul cd ĉ d ĉ c, (2.6) abcd where V coul is the Coulomb potential and E a the energy of the non-interacting MES defined by E a = i n i E i, (2.7) where E i is the energy of the i-th single-electron state. Of course (2.6) is equivalent to the original Schrödinger equation in first quantization but with this formalism the anti-symmetry of the MES wave-functions are incorporated into the Hamiltonian, making calculations much easier. 2.2 Making the MES basis At this point we have already calculated the SES s, so the task of making the the basis is only a book-keeping problem. This is best explained with an example. Suppose we have two electrons and 5 single-electron states. There are ( 5 2) = 10 ways of ordering these two electrons (the electrons are indistinguishable). An example how to order them is shown in table 1 The pattern is obvious and is easy to generalize to a MES with N SES singleelectron states. A similar method can be used for more than two electrons but the number of many-electron states N MES grows very fast with increasing number of electrons according to (2.8) ( ) NSES N MES =, (2.8) N e 2

5 Table 1: Many-electron states for N e = 2 and N SES = 5 # State where N e is the number of electrons. This can be seen in table 2. There is one problem with this method of making the MES s. The energy of state number i is not generally lower than that of state number i + 1. We can, however, number the states any way we like after the energy of each state has been determined. All that matters is that we have a consistent method of making all possible states. Table 2: N MES for some values of N e and N SES N e \ N SES Calculating the matrix elements So far we have denoted the single-electron states as i, in order to make a distinction between many- and single-electron states we denote many-electron states by i). Now we need the matrix elements H µν = abcd ab V coul cd (µ ĉ aĉ bĉdĉ c ν). (2.9) The right term is easy to calculate using the properties of the Fermi operators in (2.2) through (2.4). For the left term we need to do a 4-dimensional integral [2] ab V coul cd = dr dr Ψ a(r)ψ b(r )V coul (r r )Ψ c (r )Ψ d (r), (2.10) 3

6 where Ψ i (r) is the SES wave-function of state i and V Coul (r r ) = e 2 κ (x x ) 2 + (y y ) 2 + η, (2.11) with e 2 κ = Ω ω a ω a 0, a 0 = 2 κ = 9.79 nm. (2.12) m e2 It is convenient to write ab V coul cd = drψ a(r)i bc (r)ψ d (r), (2.13) where I bc (r) = dr Ψ b(r )V coul (r r )Ψ c (r ). (2.14) The η in (2.11) is a positive real number who purpose is to prevent the singularity when r = r. To find a suitable value for η we need to experiment a bit with it. This will be done later. 2.4 Calculating the charge density One of the most interesting observables in our system is the charge density Q s. To calculate it I use ˆQ s (x) = e φ aφ b ĉ aĉ b ˆQ s (x) = e φ aφ b µ ˆρĉ a ĉ b µ nm nm µ = e ab φ aφ b ρ µν ν ĉ a ĉ b µ, (2.15) µν where the φ i are single-electron states. 2.5 Bottlenecks in calculations One of the bottlenecks in the calculations is the right term on the right side of (2.6) we have to do a quadruple sum for N 2 mes matrix elements so the time it takes to compute the sum (given that all the integrals ab V coul cd have been calculated) is proportional to N 2 MESN 4 SES = ( NSES N e ) 2 N 4 SES. (2.16) 4

7 Another bottleneck is the 4-dimensional integral ab V cd, which we have to do for all combinations of a,b,c and d. The integral is split into two parts, (2.13) and (2.14) of which (2.14) is is much more expensive to calculate with a reasonable choice of grid size and N SES. The time it takes to compute I bc (r) for all combinations of b and c is proportional to N 2 SESN 2 xn 2 y, (2.17) where N x and N y are the number of points in the x- and y-grid. This means that if we double the amount of points in both x- and y-grid, the execution time is multiplied by 16. We will later investigate how the grid size effects the accuracy of the calculations. The biggest bottleneck is calculating the charge density (see eq. (2.15)). The computation amount is proportional to ( ) 2 NSES N x N y NSES 2. (2.18) N e Luckily we are only interested in plotting the lowest states and ρ µν 0 for large µ and ν so work can be saved without losing much precision by only summing over the first µ and ν. It is important to find a good balance between the choice of N SES and the grid size. For example there is no point in having N SES huge if the error introduced because of the finite grid is much larger than the error due to the limited N SES and vice versa. 2.6 The program The program for these calculations works exactly the same as for the non-interacting case until after the single-electron states with the effects of the magnetic field and geometry of the system have been calculated. Here, I will therefore just explain the part of the program that is used only in the interacting case. I will begin listing the functions and subroutines used and explain their function. Fermi4CStructure: This function calculates (µ ĉ aĉ bĉdĉ c ν). It has input parameters µ, ν, a, b, c, d. Implementing this as a function is efficient because each combination of the input parameters is only used once, so saving the values to a rank 6 matrix would both be pointless and impossible due to finite RAM. Inside this function I have defined two internal functions (the Fermi destruction and creation operators). make_coef: This subroutine uses the method explained in table 1 to make the many-electron states for two electrons. The states are saved in a matrix 5

8 called coef whose rows are many-electron states. The rows are of length N SES + 2. The two extra elements are the sign of the state (element number 1) and existence (element number 0). If existance = 0 in a MES that means a raising operator has been applied to an occupied state or destruction operator to an empty state. make_coef3: Same as make_coef except it creates the states for 3 electrons. MatElCoul: This subroutine has input parameters V Coul and η. It calculates the 4-dimensional Coulomb integral (2.13) for all values of a,b,c and d and saves it to the rank 4 matrix abvcd. CoulInt: This subroutine calculates the quadruple sum in (2.9). It must be called after MetElCoul because it uses its result. make_gamma: This subroutines makes a rank 2 matrix whose elements gamma(i,k) = γ k for state i), see (2.5). This matrix is used when calculating the charge density. Del_Qn2: This subroutine uses (2.15) to compute and write to a file the charge density of a chosen number of many-electron states with and without e-e interaction. 3 Convergence Like mentioned in section 1, no theoretical approximation has been done. In this section we will investigate how the finite MES basis (limited N SES ) and grid size affects the accuracy of the calculations. We will also investigate the effects of varying the size of η in (2.11). When one of those parameters is varied, all the others will be held constant. 3.1 Effects of limited MES basis If we have N MES many-electron states in our basis we will obtain N MES energy (eigen) values from the calculations. The accuracy is highest for the ground state and decreases as we go up in energy. This is a very convenient behavior for us because we are only interested in the first 200 or so excited states. I ran the program for two electrons and varied the value of N SES (see figure 1). To get an idea of the precision of the first 200 states I plotted the relative difference between the energy spectrum of the N SES = 48 case and all the other values, that is E 48 E 40 E 4, E 48 E 32 0 E 4 etc. (see figure 2). From this data we can estimate that the error is at most 0.6% if we use N ses = 48 for the first 200 states, 8 less than 0.01% for the first 20 states and less than 0.001% for the ground state. 6

9 Figure 1: Comparison of the energy spectrum for a few values of N SES. The energy spectrum of the non-interacting basis is also plotted for comparison. Although the energy spectrum is discrete, I drew lines between the values for easier comparison. 3.2 Effects of limited grid The effects of the limited grid come into play when doing the Gaussian integration. Unlike the error due to the limited MES-basis, the grid error affects all the states equally. We would therefore definitely not want the error to be higher than 0.6% (that s the maximum error due to the limited basis) but we also don t want to lose precision on the ground state so we are not gonna settle with error of more than 0.001%. I ran the program for varying grid size namely 80x40 (grid 1), 160x80 (grid 2), 320x160 (grid 3), 480x240 (grid 4) and 640x320 (grid 5) and plotted the relative difference between the resulting energies from grid 5 and all the other ones (see figure 3). From the data we see that grid 2 is sufficiently large but the difference is computational time for grid 2 and grid 3 is very small so we might as well use grid 3 (calculating the matrix elements is still a much more expensive procedure for these grid sizes). 3.3 Choosing a suitable η The effects of the choice of η are a little more complicated to assess, for, in my experience it is also highly dependent on grid size and therefore requires special 7

10 (a) N ses = 16 (b) N ses = 24 (c) N ses = 32 (d) N ses = 40 Figure 2: Relative error plotted as a function of state number for multiple values of N ses (a) Grid 1 (b) Grid 2 (c) Grid 3 (d) Grid 4 Figure 3: Relative error plotted as a function of state number for multiple grids. 8

11 attention. I ran my program for 5 values of η (0.1, 0.01, 0.001, and ) and two grid sizes (grid 2 and grid 4). Like before I will calculate the relative difference of the obtained energy values but I will begin by investigating how the choice of η affects the integral (2.14) by plotting it for a chosen combination of b and c (see figure 4). For the smaller grid we start seeing some irregularities in I bc when η is 10 4 or less while it looks fine with the bigger grid. These irregularities I call η-artifacts. These η-artifacts don t start showing up on the bigger grid until η 10 5 and even then the artifacts are barely noticeable. These artifacts obviously decrease with bigger grid but unfortunately grid 4 is as big as we can have it without drasticly increasing the program execution time. I therefore conclude that the best choice of η is 10 4 along with grid 4. Now we investigate the convergence of the energy spectrum with decreasing η. We saw earlier that η = 10 4 with grid 4 gave the best resulting I bc so I will calculate and plot E η=10 4 E η=ηi E (3.1) η=10 4 for η i = 10 1, 10 2 and 10 3 (see figure 5). These results show that the error due to the finite η is less than 0.001%, which is not much, but it it still the largest cause of error. 3.4 Conclusion on convergence We have investigated the convergence of the program s results for all our numerical approximations and the results were positive (otherwise the program would be worthless). We found that we have at least 5 correct significant figures on our energy spectrum (0.001%relative error). Getting better precision would require us to use a bigger grid which is not practical as we preferably want the calculations to be over in our lifetime. 9

12 (a) η = 10 2, grid 2 (b) η = 10 2, grid 4 (c) η = 10 3, grid 2 (d) η = 10 3, grid 4 (e) η = 10 4, grid 2 (f) η = 10 4, grid 4 (g) η = 10 5, grid 2 (h) η = 10 5, grid 4 Figure 4: Ibc plotted with a = 1, b = 2 for a few η s. 10

13 (a) η i = 10 1 (b) η i = 10 2 (c) η i = 10 3 Figure 5: Relative error plotted as a function of state number for a few values of η i. 11

14 4 Physical results Now that we have gotten all the boring (but necessary) convergence calculations out of the way we can finally start to look at the physically relevant data. We will start by looking at how the e-e interaction affects the energy spectrum, especially the energetically lowest 10 states. We will also investigate how the charge density changes and compare it with the non-interacting case. 4.1 Energy spectrum General behavior We will start by investigating the energy spectrum of the lowest 10 states for a broad range of magnetic field and comparing the results with the non-interacting case. I did the calculations for a magnetic field in the range T in increments of 0.1 up to 1.0 T and 0.5 up to 4.0 T. I used N SES = 40, η = 10 4 and the grid size was Each run took about 40 hours on Sol, running on 8 cores (practically) the entire time. On figure 6 we can see the effects of the magnetic field on the energy spectrum. Without the magnetic field the energy increases very evenly (exept for the first two states), but with increasing magnetic field the states tend to group into groups with similar energy. The energy spectrum begins to look more like a step function, rather than continuous. This effect does not show up in the non-interacting case. As expected, including the interaction raises the energy by about 1.5meV. To see better the the grouping of the energy states I plotted the energy of the 20 lowest states as a function of magnetic field (see figure 7). On figure 7a we can clearly see how the states tend to group together in higher magnetic field while without the interaction (figure 7b) we see no such effects Effects of magnetic field on lowest states Now let s investigate the effects of the magnetic field on the lowest states by plotting the energy of state n as a function of magnetic field. The results are in figure 8. As we can see from the figure the energy increases with increasing magnetic field, as expected (we are pumping energy into the system by applying a magnetic field to it). We also see that the separation in the energy spectrum between the interacting and non-interacting case stays practicly constant with increasing magnetic field but the energy of the states increases. This means the interaction plays a lesser role with increasing magnetic field. For example the seperation of the ground state for B = 0.0 T is about 90% but only 14% for B = 4.0 T. 12

15 (a) B = 0.0 T (b) B = 0.2 T (c) B = 0.4 T (d) B = 0.8 T (e) B = 1.0 T (f) B = 2.0 T Figure 6: Comparison of the energy of the 10 lowest states with the non-interacting case. With interaction is in red and without in blue. 13

16 (a) With interaction. (b) Without interaction. Figure 7: Energy of the 20 lowest states as a function of magnetic field with and without interaction. 14

17 (a) Ground state (state 1) (b) First excited state (state 2) Figure 8: Comparison of the energetically lowest 3 states as a function of magnetic field. With interaction is in red and without in blue. 15

18 (c) Second excited state (state 3) (d) Third excited state (state 4) Figure 8: Comparison of the energeticly lowest 3 states as a function of magnetic field. With interaction is in red and without in blue. 16

19 4.1.3 Energy splitting Now let s investigate how the system responds to excitation by calculating how much energy is required to excite the system from a state to a higher one. The higher this energy is, the harder it is to excite the system. I did this by calculating the difference in energy between a few selected states as a function of magnetic field, that is 1 = E 2 E 1, 2 = E 3 E 2, 3 = E 4 E 3. (4.1) See figure 9. From figure 9a we see that for B 0.25 T, 1 is biggest and then decreases with increasing magnetic field, converging at around 0.2 mev. The effect is similar for the non interacting case, but the energy is shifted. From figure 9b we see a different behaviour. There is a maximum of 2 for no magnetic field and then it decreases rapidly and goes very close to zero. This means the states E 2 and E 3 are close to being degenerate for large B. (This grouping of energy states has been mentioned earlier in this section). This effect is not apparent without the interaction. For 3 we see a similar behaviour as for 1, there is a maximum around 0.5 T and then 3 converges to around 0.15 mev while the behaviour without the interaction is very different. 17

20 (a) E 2 E 1 (b) E 3 E 2 (c) E 4 E 3 Figure 9: Energy splitting as a function of magnetic field. Red is for the interacting case and blue for the non-interacting one 18

21 4.2 Charge density We will start by looking at the charge density of the ground state for a few values of magnetic field (see figure 10). From the figure we see that the charge density is not strongly affected by the magnetic field as long it is lower than 0.5 T. In strong magnetic field the charge density tends to smear out in the ±y-direction (forming of edge states). Let s now how the first and second excited states are affected (see figures 11 and 12). From these figures we can see that the effects of the interaction is much greater, especially in higher magnetic field. The first and second excited states are not greatly modified in zero magnetic field but they is distinctively different in a modest 0.2 T magnetic field. The hump in the middle of figure 11d disappears when the interaction is included. This is because of the electron electron repulsion which makes areas of high charge density energetically unfavorable. The magnetic field seems to have very different effects on the interacting case compared to the non-interacting one. For examples on figure 12 we see that without the magnetic field both the interacting and non-interacting charge densities have areas of high density. With increasing magnetic field the interacting density seems to smear out and spread more uniformly while the non-interacting case retains its areas of high density. 5 Conclusions We have used numerical techniques to calculate the energy spectrum and charge distribution of two electrons in a 200nm GaAs wire in a constant homogeneous magnetic field including the Coulomb interaction between the electrons and compared the results with the non-interacting case. According to section 3 the results converged very nicely when the basis/grid was enlarged and the biggest cause of error was the non-zero value of η (see section 3.3). In section 4 we saw how the lowest states are not greatly affected by the magnetic field nor the Coulomb interaction but when we go higher in energy the system is highly sensitive to the interaction and magnetic field. There is a lot more to be studied in this system. One example is adding the third electron. The program is capable of doing the calculations for three electrons (and it is easy to generalize it to any number of electrons) but the basis has to be shrunk considerably (see table 2) so calculations beyond 4-7 electrons are not very practical. Another example is adding a time dependent perturbation which would force us to truncate the many-electron basis from a few thousand states to a few hundred to make numerical calculations possible. 19

22 (a) With interaction, B = 0.0 T (b) Without interaction, B = 0.0 T (c) With interaction, B = 0.5 T (d) Without interaction, B = 0.5 T (e) With interaction, B = 2.0 T (f) Without interaction, B = 2.0 T (g) With interaction, B = 4.0 T (h) Without interaction, B = 4.0 T Figure 10: Plot of charge density of the ground state for a few values of magnetic field, with and without interaction. 20

23 (a) With interaction, B = 0.0 T (b) Without interaction, B = 0.0 T (c) With interaction, B = 0.2 T (d) Without interaction, B = 0.2 T (e) With interaction, B = 0.5 T (f) Without interaction, B = 0.5 T (g) With interaction, B = 4.0 T (h) Without interaction, B = 4.0 T Figure 11: Plot of the charge density of the first excited state for a few values of magnetic field, with and without interaction. 21

24 (a) With interaction, B = 0.0 T (b) Without interaction, B = 0.0 T (c) With interaction, B = 0.2 T (d) Without interaction, B = 0.2 T (e) With interaction, B = 0.5 T (f) Without interaction, B = 0.5 T (g) With interaction, B = 4.0 T (h) Without interaction, B = 4.0 T Figure 12: Plot of the charge density of the second excited state for a few values of magnetic field, with and without interaction. 22

25 References [1] Fetter and Walecka. Quantum Theory of Many-particle Systems, pages McGRAW-HILL BOOK COMPANY, [2] Viðar Guðmundsson. Gme, interaction in the system,